On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H
Abstract
The Markov kernel based metric D1 was introduced in 2011 in order to construct the scale-invariant dependence measure ζ1, which assign each bivariate copula C a dependence value in [0,1], with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space (𝒞,D1) is separable and complete, however, no further topological properties were studied. Considering that D1 has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that (𝒞,D1) is homeomorphic to the Hilbert space (ℓ2,‖ · ‖2), and prove that several subfamilies are either homeomorphic to (ℓ2,‖ · ‖2) or to the Hilbert cube (ℋ,ρ). Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called Z-sets in (𝒞,D1), implying that they are topologically negligible in the full space.
Related
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- Numerical Information of Hilbert Space Operators
BibTeX
@misc{sanchez2026families,
title = {{On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H}},
author = {Juan Fernández Sánchez and Wolfgang Trutschnig},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29275},
eprint = {2609.29275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29275},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H
AU - Juan Fernández Sánchez
AU - Wolfgang Trutschnig
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29275
UR - https://arxiv.org/abs/2609.29275
AB - The Markov kernel based metric D_1 was introduced in 2011 in order to construct the scale-invariant dependence measure ζ_1, which assign each bivariate copula C a dependence value in [0,1], with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space (𝒞,D_1) is separable and complete, however, no further topological properties were studied. Considering that D_1 has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that (𝒞,D_1) is homeomorphic to the Hilbert space (ℓ_2,‖ · ‖_2), and prove that several subfamilies are either homeomorphic to (ℓ_2,‖ · ‖_2) or to the Hilbert cube (ℋ,ρ). Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called Z-sets in (𝒞,D_1), implying that they are topologically negligible in the full space.
ER -